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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.19487 |
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| _version_ | 1866918259317538816 |
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| author | Saurabh, Prasoon |
| author_facet | Saurabh, Prasoon |
| contents | The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators $\mathcal{L}(k)$ over a complex parameter manifold $X$ canonically defines a \textbf{regular holonomic $\mathcal{D}_X$-module} $\mathcal{M}$. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the \textbf{Kashiwara filtration}, we show that the open quantum system underlies a \textbf{Mixed Hodge Module (MHM)} structure in the sense of Saito.
This identification allows us to apply the \textbf{Grothendieck six-functor formalism} rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in $\text{Ext}^1_{\mathcal{D}_X}$, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the ``singular component'' of the Complete QGT arises as the residue of the connection on the \textbf{Brieskorn lattice} associated with the vanishing cycles functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities Saurabh, Prasoon Mathematical Physics Algebraic Geometry Quantum Physics 32C38, 32S35, 32S40, 81Q12, 81S22 J.2; G.0; I.1.4; F.4.1 The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators $\mathcal{L}(k)$ over a complex parameter manifold $X$ canonically defines a \textbf{regular holonomic $\mathcal{D}_X$-module} $\mathcal{M}$. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the \textbf{Kashiwara filtration}, we show that the open quantum system underlies a \textbf{Mixed Hodge Module (MHM)} structure in the sense of Saito. This identification allows us to apply the \textbf{Grothendieck six-functor formalism} rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in $\text{Ext}^1_{\mathcal{D}_X}$, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the ``singular component'' of the Complete QGT arises as the residue of the connection on the \textbf{Brieskorn lattice} associated with the vanishing cycles functor. |
| title | Open Quantum Systems as Regular Holonomic $\mathcal{D}$-Modules: The Mixed Hodge Structure of Spectral Singularities |
| topic | Mathematical Physics Algebraic Geometry Quantum Physics 32C38, 32S35, 32S40, 81Q12, 81S22 J.2; G.0; I.1.4; F.4.1 |
| url | https://arxiv.org/abs/2512.19487 |